Abstract :
The double angle theorems of Davis and Kahan bound the change in an invariant subspace when a Hermitian matrix A is subject to an additive perturbation A→Ã=A+ΔA. This paper supplies analogous results when A is subject to a congruential, or multiplicative, perturbation A→Ã=D*AD. The relative gaps that appear in the bounds involve the spectrum of only one matrix, either A or Ã, in contrast to the gaps that appear in the single angle bounds.
The double angle theorems do not directly bound the difference between the old invariant subspace
image
and the new one
image
but instead bound the difference between
image
and its reflection
image
where the mirror is
image
and J reverses
image
, the orthogonal complement of
image
. The double angle bounds are proportional to the departure from the identity and from orthogonality of the matrix
image
. Note that
image
is invariant under the transformation D→D/α for α≠0, whereas the single angle theorems give bounds proportional to Dʹs departure from the identity and from orthogonality.
The corresponding results for the singular value problem when a (nonsquare) matrix B is perturbed to
image
are also presented.
